Ray optics all derivations

RAY OPTICS – ALL DERIVATIONS Derivation of mirror formula Consider an object AB placed on the principle axis beyond the centre of curvature C of a concave mirror of small aperture, as shown. Now A 'B'C ABC A 'B' CB' CP B'P R v .........(i) AB BC BP CP u R As A 'B'C APB, therefore, A 'B'P ABP Consequently, A 'B' B'P v v .................(ii) AB BP u u From(i) and (ii),we get R v v u R u uR uv uv vR vR uR                                      2uv vR uR 2uv uvR uvR uvR 1 1 2 u v R As R 2f 1 1 2 so u v 2f 1 1 1 u v f            ___________________________________________________________________ Relation between focal length and radius of curvature for spherical mirror From the diagram, As AB is parallel to PC, α i In BFC, r α Hence CF FB For a mirror small aperture FB FP CF FP hence CP CF FP FP FP 2FP or R 2f                  ___________________________________________________________________ Derivation of thin lens formula Consider and object AB placed perpendicular to the principal axis of a thin convex lens between its F’ and C’ as shown. A real, inverted and magnified image A’B’ is formed beyond C on the other side of the lens. A 'B'O and ABO are similar A 'B' OB' (1) AB BO Also A 'B'F and MOF are similar A 'B' FB' MO OF But MO AB A 'B' FB' .......(2) AB OF From (1) and (2), we get OB' FB' OB' OF BO OF OF By sign cartesian sign convention, we get Object distance, B               O = -u Image distance, OB' = +v Focallength OF f v v f u f or vf uv uf or uv uf vf Dividing both sides by uvf, we get 1 1 1 f v u              ___________________________________________________________________ Derive a relation between critical angle and refractive index of a medium. Consider a light ray travelling from denser medium (b) to a rarer medium a. According to Snell’s law, a a sini μ sinr  Where, b represents denser medium to rarer medium. o c b c a c o At i i , r 90 sini μ sini sin90    b a a b a b c 1 But μ μ Therefore 1 μ sini   _________________________________________________________________________ Derive the relation between the distance of object, distance of image and radius of curvature of convex spherical surface, when refraction takes place from rarer to denser medium and image formed is real. Consider an object placed at O and its real image is formed at I as shown. Similarly In NOC, i is an exterior angle, therefore, i α γ , from NIC, we have γ r β r γ β          Suppose, all the rays are paraxial. Then the angles i, r, α, β and γ will be small. NM NM α tanα OM OP NM NM β tanβ MI PI NM NM γ tanγ MC PC           From Snell’s law of refraction, 2 1 sini μ sinr μ As i and r are small, therefore 2 1 i μ r μ      1 2 1 2 1 2 1 2 1 2 2 1 μ i μ r μ α γ μ γ β NM NM NM NM μ μ OP PC PC PI 1 1 1 1 μ μ OP PC PC PI μ μ μ μ OP PI PC                                           Using Cartesian sign convention, Object distance OP u   Image distance PI v   Radius of curvature PC = +R μ μ μ μ 1 2 2 1 u v R      ________________________________________________________________________ Derive the expression for lens maker’s formula. Consider an object placed at O whose final image is formed at I as shown. Let the image formed by first surface is at I1. This image will act as on abject for the second surface. For refraction at first surface, we have   2 1 2 1 1 1 1 2 1 2 1 2 1 1 2 1 1 2 2 1 1 1 2 μ μ μ μ (i) v u R for refraction at second surface we have μ μ μ μ (ii) v v R adding (i) and (ii) we get μ μ 1 1 μ μ v u R R 1 1 1 1 μ μ v u μ R R                                  If object is placed at infinity (u   ), the image is formed at focus, i.e. v = f. Therefore, 2 1 1 1 2 1 1 1 μ μ f μ R R                  2 1 1 2 1 2 1 2 1 1 1 μ 1 f μ R R 1 1 1 μ 1 f R R                           This result is lens maker’s formula. Derive a relation between angle of deviation, angle of prism and refractive index of prism. Consider a ray PQ incident of one face of a prism as shown. The path of ray inside the prism and refracted ray is also shown. From quadrilateral AQNR o o A QNR 180 From the triangle QNR r r ' QNR 180 A r ' r           Now, from the triangle MQR, the deviation produced by the prism         δ MQR MRQ i r i' r ' or δ i i' r r ' or δ i i' A or i i' A δ                   For refraction at face AB, we have sini i μ i μr sinr r     For refraction at face AC, we have sini' i' μ i' μr ' sinr ' r '     Hence deviation produced by the prism is     δ i i' A μr μr ' A δ μ r r ' A μA A δ μ 1 A                ________________________________________________________________________ Derive prism formula Or Derive a relation for refractive index of a prism in terms of angle of minimum deviation. When a prism is in the position of minimum deviation, a ray of light passes symmetrically (parallel to base) through the prism, so that m i i', r r ', δ δ    As m m A δ i i' A δ A δ i i' or i 2 Also A r r ' r r 2r A r 2                 From Snell’s law, the refractive index of the material of the prism will be A δm sin sini 2 μ or μ sinr A sin 2                _________________________________________________________________________ Derive an expression for magnifying power of a simple microscope when final image in formed at a. Least distance of distinct vision. b. Infinity When final image is formed at least distance of distinct vision The image A’B’ of an object AB is formed at least distance of distinct vision ‘D’ as shown. Let A 'OB' β   . Imagine the object AB to be placed to position A ''B ' at distance D from the lens. Let   A''OB' α . Then, magnifying power, β tanβ m α tanα   [since α and β are small] AB / OB AB / OB [ A ''B' AB] A ''B'/ OB' AB / OB' OB' D OB x D or m x          Let f be the focal length of the lens. As the image is formed at least distance of distinct vision from the lens, so v D   Using thin lens formula, 1 1 1 v u f we get 1 1 1 D x f 1 1 1 x D x D D 1 x f D m 1 f                When final image is formed at infinity From fig (a) h tanβ f  From fig (b) h tanα D h / f m h / D D or m f      _________________________________________________________________________ Derive an expression for magnifying power of a compound microscope when final image in formed at a. Least distance of distinct vision. b. Infinity When final image is formed at least distance of distinct vision The object AB is placed at uo slightly larger than the focal length o f of the objective O. The object forms a real, inverted and magnified image A’B’ on the other side of the lens. This image acts as an object for the eyepiece which essentially acts like a simple microscope. The eyepiece E forms a virtual and magnified final image A’’B’’ of the object.